Categories Determined by Inverse Semigroups · · 2016-07-28One category of interest in studying...
Transcript of Categories Determined by Inverse Semigroups · · 2016-07-28One category of interest in studying...
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Categories Determined by Inverse
Semigroups
A. R. Rajan1
University of Kerala, Thiruvananthapuram
Kerala 695 581, India
1The author thanks Kerala State Council for Science Technology and
Environment for support under Emeritus Scientist SchemeA. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Introduction
Several categories arise in the structure theory of inverse semigroups
and also of regular semigroups.
For example the well known structure theorem of Schein(cf. [10])
for inverse semigroups is based on the inductive groupoid associated
with the inverse semigroup. Here a groupoid is a category in which
all morphisms are isomorphisms.
Several other categories associated with inverse semigroups can be
seen in literature, for example J.E. Pin and Stuart Margolis [6], M.V.
Lawson[3], AR Rajan[8] etc.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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One category of interest in studying structure of regular semigroups
is the category of principal left [right] ideals of the semigroup intro-
duced by K.S.S. Nambooripad (cf.[5]).
These categories have been abstractly characterised as normal cate-
gories. Normal categories are the basic categories considered in the
cross connection theory of regular semigroups, which is a general-
ization of Grillet’s cross connection (cf.[1]).
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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One interesting question associated with study of normal categories
is regarding tne characterization of normal categories arising from
various classes of regular semigroups. We consider this question for
certain classes of inverse semigroups in this talk.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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The contents of the talk are the following.
1 Normal categories associated with regular semigroups
2 Normal categories arising from inverse semigroups
3 Normal categories from E-unitary inverse semigroups
4 Characterization of normal categories arising fron Clifford
semigroups
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Normal Categories from Regular Semigroups
The normal category L(S) of principal left ideals of a regular semi-
group S is described by Nambooripad (cf.[5]) as follows.
The objects of L(S) are principal left ideals Sa with a ∈ S.
Since S is a regular semigroup every principal left[right] ideal is
generated by an idempotent. So vL(S) = {Se : e ∈ E(S)} where
E(S) is the set of all idempotents of S.
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For Se, Sf in vL(S), a morphism from Se to Sf is a map
ρ(e, v, f) : Se→ Sf
which maps x ∈ Se to xv ∈ Sf where v ∈ eSf.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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It may be noted that eρ(e, v, f) = v and corresponding to each
v ∈ eSf there is a unique morphism ρ(e, v, f) : Se→ Sf .
For morphisms ρ(e, u, f) : Se → Sf and ρ(f, v, h) : Sf → Sh the
product is defined as the composition of the given maps and this
becomes ρ(e, uv, h) : Se→ Sh.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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A normal category is abstractly described as follows.
The description begins with the concept of a category with subob-
jects which is a pair (C,P) where C is a small category and P is a
subcategory of C in such a way that vP = vC and P is a strict
preorder.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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In this case vC = vP becomes a partially ordered set with respect
to the preorder structure of vP.That is for a, b ∈ vC
a ≤ b if there is a morphism from a to b in P.
The morphisms of P are called inclusions and are denoted as
j(a, b) : a→ b.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Another concept used in the description of normal category is that
of normal cone.
A normal cone in a category with subobjects (C,P) is a map
γ : vC → C
such that
1 there is an object c such that for every a ∈ vC, γ(a) : a→ c.
This c is called the vertex of γ and is usually denoted by cγ
2 if a, b ∈ vC are such that a ≤ b then γ(a) = j(a, b)γ(b)
3 There is at least one a ∈ vC such that γ(a) is an isomorphism.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Definition 2.1
A category with sub objects (C,P) is said to be a normal category
if the following hold.
(1.) Every inclusion j = j(a, b) : a→ b has a right inverse
q : b→ a in C. Such a morphism q is called a retraction in C.(2) Every morphism f ∈ C has a factorization in the form
f = quj
where q is a retraction, u is an isomorphism and j is an
inclusion. Such a factorization is called a normal factorization.
(3) For each c ∈ vC there exists a normal cone γ such that
γ(c) = 1c.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Each normal category C gives rise to a regular semigroup denoted by
TC, which is the semigroup of all normal cones in C with a suitable
composition.
Moreover the normal category C is isomorphic to the normal category
L(S) where S = TC.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Inverse Semigroups
Now we consider the special properties of the normal category L(S)when S is an inverse semigroup.
The following theorem describes the major properties of the normal
category L(S) of an inverse semigroup S.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Theorem 3.1
Let S be an inverse semigroup and L(S) be the normal category of
principal left ideals of S. Then
1 Every morphism ρ(e, u, f) : Se→ Sf in L(S) has a unique
normal factorization.
2 If Se ⊆ Sf then there is unique retraction from Sf to Se.
3 Let j = ρ(e, e, f) : Se→ Sf be an inclusion in L(S) and
q = ρ(f, g, g) : Sf → Sg be a retraction in L(S) such that jq
is an isomorphism. Then e = g and jq is identity.
4 For each object Se ∈ L(S) there is a normal cone γ in L(S)whose vertex is Se and γ(Se) = 1Se.
5 For each a ∈ S let ρa be the cone with vertex Se = Sa and
components ρa(Sg) = ρ(g, ga, e). Then ρa is a normal cone in
L(S) and ρa has exactly one isomorphism component.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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E-Unitary Inverse Semigroups
It may be noted that all normal cones in L(S) are not of the form
ρa for some a ∈ S.We now show that if S is an E−unitary inverse semigroup then
idempotent normal cones in L(S) are of the form ρe and L(S) has
other nice properties.
An inverse semigroup S is said to be E−unitary if for e ∈ E(S) and
s ∈ S if es ∈ E(S) or se ∈ E(S) then s ∈ E(S) (cf[2]).
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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We have the following special properties for L(S) in this case.
Theorem 3.2
Let S be an E−unitary inverse semigroup, e ∈ E(S) and γ be a
normal cone in L(S) with vertex Se such that γ(Se) = 1Se. Then
γ = ρe.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Another property of the normal category arising from E-unitary in-
verse semigroups is the following.
Theorem 3.3
Let S be an E−unitary inverse semigroup and γ be a normal cone
in L(S). Then if γ(a) and γ(b) are isomorphisms then a = b.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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E-Unitary Clifford Semigroups
Another class of inverse semigroups for which the associated normal
category has a simple characterization is the class of E-uniary Clifford
semigroups.
Clifford semigroups are inverse semigroups which are semilattices of
groups.
The following theorem characterises normal categories determined
by E-unitary Clifford semigroups.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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Theorem 3.4
Let C be a normal category such that vC is a semilattice with
respect to the the partial order given by the inclusion in Csatisfying the following.
1 Normal factorizations and retractions are unique.
2 A morphism f : a→ b is an isomorphism if and only if a = b.
3 If a ≤ b then for every u ∈ C(a, b) the map u 7→ j(a, b)uq(b, a)
is a one to one map from C(b, b) to C(a, a).4 Let j : a→ b be an inclusion and q : b→ c be a retraction in
C then jq = q1j1 for some retraction q1 and inclusion j1.
Then the semigroup of normal cones S = TC is an E-unitary
Clifford semigroup and C is isomorphic to L(S). Conversely if S is
any E−unitary Clifford semigroup then L(S) satisfies the
conditions above.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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References I
[1] P. A. Grillet, Structure of regular semigroups, I. A
representation; II Cross-connections; III The reduced case,
Semigroup Forum 8(1974), 177-183, 254-265.
[2] J.M. Howie, Fundamentals of Semigroup Theory, Academic
Press, New York, 1995.
[3] M.V. Lawson, Semigroups and ordered categories I, The
reduced case, J. Algebra 141(1991),422-462.
[4] S. Mac Lane, Categories for the Working Mathematician,
Springer-Verlag, New York, 1971.
[5] K.S.S. Nambooripad, Theory of Cross Connections, Publ. No.
38, Centre for Mathematical Sciences, Trivandrum, 1984.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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References II
[6] J.E. Pin and Stuart Margolis, Inverse semigroups and
extensions of graphs by semilattices, J. Algebra
110(1987),277–297.
[7] A. R. Rajan, Ordered groupoids and normal categories, Proc.
Int. Conference in Semigroups and its related topics. Editors
K. P. Shum and others, Springer 1995.
[8] Rajan, A.R., Normal categories of inverse semigroups, East
West J. Math. 16(2) (2014) 122–130.
[9] A. R. Rajan, Certain categories derived from normal
categories, Proc. Int. Conference on Semigroups, Algebras
and Operator Theory, Editors P.G. Romeo, John C. Meakin
and A.R. Rajan Springer 2015.
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups
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References III
[10] Schein, Boris, On the theory of generalised groups and
generalised heaps, University of Saratov, 1965, 286–324
(Russian).
A. R. Rajan University of Kerala, Thiruvananthapuram Kerala 695 581, India [email protected] Determined by Inverse Semigroups